Generalized Schr\"oder paths arising from a combinatorial interpretation of generalized Laurent bi-orthogonal polynomials

Lattice paths called $\ell$-Schr\"oder paths are introduced. They are paths on the upper half-plane consisting of $\ell+2$ types of steps: $(i,\ell-i)$ for $i=0,\ldots,\ell$, and $(1,-1)$. Those paths generalize Schr\"oder paths and some variants, such as $m$-Schr\"oder paths by Yang and Jiang and Motzkin-Schr\"oder paths by Kim and Stanton. We show that $\ell$-Schr\"oder paths arise naturally from a combinatorial interpretation of the moments of generalized Laurent bi-orthogonal polynomials introduced by Wang, Chang, and Yue. We also show that some generating functions of non-intersecting $\ell$-Schr\"oder paths can be factorized in closed forms.